The Parameterized Complexity of Finding Minimum Bounded Chains
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Finding the smallest d-chain with a specific (d-1)-boundary in a simplicial complex is known as the Minimum Bounded Chain (MBC_d) problem. The MBC_d problem is NP-hard for all d≥ 2. In this paper, we prove that it is also W[1]-hard for all d≥ 2, if we parameterize the problem by solution size. We also give an algorithm solving the MBC_1 problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC_d problem for all d. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.
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